(y, V) = , 0) . (I) This is practically a special case o f (I) i n Theorem 9; i t i s only necessary to observe that when k = In, it is clearly unnecessary, in the proof of this case, to permute the yi in order to arrange that PROOF.

PROOF. We can assume that A is contained in a compact set C (since �n is a countable union of compact sets). Lemma 5 implies that there is some K such that IJ(x) - J(y ) 1 ::; n 2 Klx - yl for all x, y E C. Thus J takes rectangles of diameter d into sets of diameter ::; n2 K d. This clearly implies that J(A) has measure ° if A does . •:. A subset A of a Coo n-manifold M has measure zero if there is a sequence of charts (XI, Vi), with A c Ui Vi, such that each set xi (A n Vi) c �" has measure 0.